Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x+3y &= 7 \\ 8x-2y &= -4\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $2$ and the bottom equation by $3$ $\begin{align*}-10x+6y &= 14\\ 24x-6y &= -12\end{align*}$ Add the top and bottom equations. $14x = 2$ Divide both sides by $14$ and reduce as necessary. $x = \dfrac{1}{7}$ Substitute $\dfrac{1}{7}$ for $x$ in the top equation. $-5( \dfrac{1}{7})+3y = 7$ $-\dfrac{5}{7}+3y = 7$ $3y = \dfrac{54}{7}$ $y = \dfrac{18}{7}$ The solution is $\enspace x = \dfrac{1}{7}, \enspace y = \dfrac{18}{7}$.